Deriving expression for resistance in terms of current density

In summary: Can we please stop this split electric field nonsense???In summary, the equation 9.42 can be obtained by substituting the line integral of j/conductivity for V in the image from Modern Electrodynamics.
  • #1
spindecide
3
0
Is there a way to obtain equation 9.42 (I is current, j is current density, and sigma is conductivity) in the following image (from Modern Electrodynamics by Andrew Zangwill, the part on electromotive force) besides using V=IR and substituting the line integral of j/conductivity for V? The aforementioned way requires that j=conductivity*E, but due to the emf, j should be equal to conductivity*[E + E'], where E' is a fictitious electric field representing the effect of any source of EMF.
Untitled.jpg
 
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  • #2
I think this equation is reasonable and I don't see any problem with it.

Circuit-30.jpg
 
  • #3
alan123hk said:
I think this equation is reasonable and I don't see any problem with it.

But how is R written as the first integral?
 
  • #4
spindecide said:
But how is R written as the first integral?
My personal opinion as follow.

Circuit-31.jpg
 
  • #5
But j is not just equal to sigma times E as the effect of the emf needs to be considered, so that j equals sigma times (E + E').
 
  • #6
A very useful equation is shown here
Circuit-33.jpg
$$ V_1-V_2=IR_{12}-\varepsilon _{12}~~~~~\Rightarrow ~~~~~V_1-V_2=IR_{12}-\int_1^2~dl\cdot~E^{'} $$ That is to say, the voltage difference in a circuit with EMF is not equal to IR, but equal to IR minus EMF. This is actually an equation very familiar and commonly used by electrical and electronics engineers when conducting circuit analysis.

It is also worth noting that $$IR_{12}=(V_1-V_2)+\varepsilon _{12}=\int_1^2~dl\cdot~(E+E^{'}),~~~~\text{so}~~R=\frac {1}{I} \int_1^2~dl\cdot~(E+E^{'}) $$
Everything is in perfect harmony, without contradictions. :smile:
 
Last edited:
  • #7
spindecide said:
But j is not just equal to sigma times E as the effect of the emf needs to be considered, so that j equals sigma times (E + E').
You are right, I accidentally wrote the equation wrong.
 
  • #8
alan123hk said:
You are right, I accidentally wrote the equation wrong.
Can we please stop this split electric field nonsense???
Please.
 
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1. What is the definition of resistance?

Resistance is a measure of how much a material or component impedes the flow of electric current through it.

2. What is current density?

Current density is the amount of electric current passing through a unit area of a material or component.

3. How is resistance related to current density?

Resistance is directly proportional to current density. This means that as current density increases, resistance also increases.

4. How can the expression for resistance be derived in terms of current density?

The expression for resistance can be derived using Ohm's law, which states that resistance is equal to the ratio of voltage to current. By substituting current density for current, the expression for resistance in terms of current density can be derived.

5. Why is it useful to have an expression for resistance in terms of current density?

Having an expression for resistance in terms of current density allows for a more precise understanding of how different materials and components affect the flow of electric current. It also allows for more accurate calculations and predictions in electrical circuits and systems.

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